jeremiahbiard

System Design Cheatsheet

Picking the right architecture = Picking the right battles + Managing trade-offs

Basic Steps

1. Clarify and agree on the scope of the system

• User cases (description of sequences of events that, taken together, lead to a system doing something useful)
  • Who is going to use it?
  • How are they going to use it?

jeremiahbiard

System Design Cheatsheet

Picking the right architecture = Picking the right battles + Managing trade-offs

Basic Steps

1. Clarify and agree on the scope of the system

• User cases (description of sequences of events that, taken together, lead to a system doing something useful)
  • Who is going to use it?
  • How are they going to use it?

jeremiahbiard

`The required answer is : 1 / (n + k) * n-choose-r(n - 3,k) * n-choose-r(n + k, n - 1).

While I am not aware of a simple way to prove this which I can describe here, here's some practical advice on how such problems can be approached. It helps to write down a simpler solution and searching Google or OEIS (Online Encyclopedia of Integer Sequences) for small inputs like k = 3 and N = 1,2,...10. This usually gives you a formula describing the sequence, and possibly generalizations for the same.

Computing the answer using the above formula requires calculating the power of MOD (1e6 + 3) in the numerator and the demonimator. If the power of MOD is bigger in the numerator, the answer is 0. Otherwise, we can calculate all the factorials with the powers of MOD cast out. For example, if MOD = 3 and N = 11, we would calculate the product 1 * 2 * 3 / 3 * 4 * 5 * (6 / 3) * 7 * 8 * (9 / 9) * 10 * 11. This can be done easily in O(MOD). Once we have that, we can multiply the values in the numerator and mltiply with the mo